On Mixed Quermassintegrals for log-concave Functions
aa r X i v : . [ m a t h . M G ] S e p ON MIXED QUERMASSINTEGRALS FOR LOG-CONCAVEFUNCTIONS
FANGWEI CHEN , JIANBO FANG , MIAO LUO , CONGLI YANG Abstract.
In this paper, the functional Quermassintegrals of log-concavefunctions in R n are discussed, we obtain the integral expression of the i -thfunctional mixed Quermassintegrals, which are similar to the integral expres-sion of the i -th Quermassintegrals of convex bodies. introduction Let K n be the set of convex bodies (compact convex subsets with nonemptyinteriors) in R n , the fundamental Brunn-Minkowski inequality for convex bodiesstates that for K, L ∈ K n , the volume of the bodies and of their Minkowski sum K + L = { x + y : x ∈ K, and y ∈ L } are given by V (cid:0) K + L ) n ≥ V ( K ) n + V ( L ) n , (1.1)with equality if and only if K and L are homothetic, namely they agree up toa translation and a dilation. Another important geometric inequality related tothe convex bodies K and L is the mixed volume, the important result concernthe mixed volume is the Minkwoski’s first inequality V ( K, L ) := 1 n lim t → + V ( K + tL ) − V ( K ) t ≥ V ( K ) n − n V ( L ) n , (1.2)for K, L ∈ K n . Specially, when choose L to be a unit ball, up to a factor, V ( K, L ) is exactly the perimeter of K , and inequality (1.2) turns out to be theisoperimetric inequality in the class of convex bodies. The mixed volume V ( K, L )admits a simple integral representation (see [31, 32]) V ( K, L ) = 1 n Z S n − h L dS K , (1.3)where h L is the support function of L and S K is the area measure of K .The Quermassintegrals W i ( K ) ( i = 0 , , · · · n ) of K , which are defined byletting W ( K ) = V n ( K ), the volume of K ; W n ( K ) = ω n , the volume of the unit Mathematics Subject Classification.
Key words and phrases.
Log-concave functions; Quermassintegral; Mixed Quermassintegral,Minkowski inequality.The work is supported in part by CNSF (Grant No. 11561012, 11861004, 11861024), GuizhouFoundation for Science and Technology (Grant No. [2019] 1055, [2019]1228), Science and tech-nology top talent support program of Guizhou Eduction Department (Grant No. [2017]069). ball B n in R n and for general i = 1 , , · · · , n − W n − i ( K ) = ω n ω i Z G i,n vol i ( K | ξ i ) dµ ( ξ i ) , (1.4)where the G i,n is the Grassmannian manifold of i -dimensional linear subspaces of R n , dµ ( ξ i ) is the normalized Haar measure on G i,n , K | ξ i denotes the orthogonalprojection of K onto the i -dimensional subspaces ξ i , and vol i is the i -dimensionalvolume on space ξ i .In the 1930s, Aleksandrov, Fenchel and Jessen (see [1, 19]) proved that fora convex body K in R n , there exists a regular Borel measure S n − − i ( K ) ( i =0 , , · · · , n −
1) on S n − , the unit sphere in R n , such that for any convex bodies K and L , the following representations hold W i ( K, L ) = 1 n − i lim t → + W i ( K + tL ) − W i ( K ) ǫ = 1 n Z S n − h L ( u ) dS n − − i ( K, u ) . (1.5)The quantity W i ( K, L ) is called the i -th mixed Quermassintegral of K and L .In the 1960s, the Minkowski addition was extended to the L p ( p ≥
1) Minkowskisum h pK + p t · L = h pK + th pL . The extension of the mixed Quermassintegrals to the L p mixed Quermassintegrals due to Lutwak [31], and the L p mixed Quermassintegralinequalities, the L p Minkowski problem are established. See [23, 32–37, 45–47] formore about the L p Minkowski theory and L p Minkowski inequalities. The firstvariation of the L p mixed Quermassintegrals are defined by W p,i ( K, L ) := pn − i lim t → + W i ( K + p t · L ) − W i ( L ) t , (1.6)for i = 0 , , · · · , n −
1. In particular, for p = 1 in (1.6), it is W i ( K, L ), and W p, ( K, L ) is denoted by V p ( K, L ), which is called the L p mixed volume of K and L . Similarly, the L p mixed Quermassintegral has the following integral represen-tation (see [31]): W p,i ( K, L ) = 1 n Z S n − h pL ( u ) dS p,i ( K, u ) . (1.7)The measure S p,i ( K, · ) is absolutely continuous with respect to S i ( K, · ), and hasRadon-Nikodym derivative dS p,i ( K, · ) dS i ( K, · ) = h K ( · ) − p . Specially, when p = 1 in (1.7)yields the representation (1.5).In most recently, the interest in the log-concave functions has been considerablyincreasing, motivated by the analogy properties between the log-concave functionsand the volume convex bodies in K n . The classical Pr´ekopa-Leindler inequality(see [13, 29, 38–40]) firstly shows the connections of the volume of convex bodiesand log-concave functions. The Blaschke-Santal´o inequality for even log-concavefunctions is established by Ball in [8, 9], the general case is proved by Artstein-Avidan, Klartag and Milman [4], other proofs are given by Fradelizi, Meyer [21]and Lehec [27, 28]. See [10, 20, 24, 44] for more about the functional Blaschke-Santal´o inequalities. The mean width for log-concave function is introduced by N MIXED QUERMASSINTEGRALS 3
Klartag, Milman and Rotem [26, 42, 43]. The affine isoperimetric inequality forlog-concave functions are proved by Artstein-Avidan, Klartag, Sch¨utt and Werner[7]. The John ellipsoid for log-concave functions have been establish by Guti´errez,Merino Jim´enez and Villa [3], the LYZ ellipsoid for log-concave functions areestablished by Fang and Zhou [18]. See [2, 6, 11, 14–16, 30] for more about thepertinent results. To establish the functional versions of inequalities and problemsfrom the points of convex geometric analysis is a new research fields.Let f = e − u , g = e − v be log-concave functions, α, β >
0, the “sum” and“scalar multiplication” of log-concave functions are defined as, α · f ⊕ β · g := e − w , where w ∗ = αu ∗ + βv ∗ , here w ∗ denotes as usual the Fenchel conjugate of the convex function ω . The totalmass integral J ( f ) of f is defined by, J ( f ) = R R n f ( x ) dx. In paper of Colesantiand Fragal`a [17], the quantity δJ ( f, g ), which is called as the first variation of J at f along g , δJ ( f, g ) = lim t → + J ( f ⊕ t · g ) − J ( f ) t , is discussed. It has been shown that δJ ( f, g ) is finite and is given by δJ ( f, g ) = Z R n v ∗ dµ ( f ) , where µ ( f ) is the measure of f on R n .Inspired by the paper of Colesanti and Fragal`a [17], in this paper, we definethe i -th functional Quermassintegrals W i ( f ) as the i -dimensional average totalmass of f , W i ( f ) := ω n ω n − i Z G n − i,n J n − i ( f ) dµ ( ξ n − i ) . Where J i ( f ) denotes the i -dimensional total mass of f defined in section four, G i,n is the Grassmannian manifold of R n and dµ ( ξ n − i ) is the normalized measureon G i,n . Moreover, we define the first variation of W i at f along g , which is W i ( f, g ) = lim t → + W i ( f ⊕ t · g ) − W i ( f ) t . It is natural extension of the Quermassintegrals of convex bodies in R n , we callit the i -th functional mixed Quermassintegral. In fact, if one takes f = χ K , and dom ( f ) = K ∈ R n , then W i ( f ) turn out to be W i ( K ), and W i ( χ K , χ L ) equals tothe W i ( K, L ). The main result in this paper is to show that the i -th functionalmixed Quermassintegrals have the following integral expressions. Theorem 1.1.
Let f , g are integrable functions on A ′ , µ i ( f ) be the i -dimensionalmeasure of f , and W i ( f, g ) ( i = 0 , , · · · , n − are the i -th functional mixedQuermassintegrals of f and g . Then W i ( f, g ) = 1 n − i Z R n h g | ξn − i dµ n − i ( f ) , (1.8) where h g is the support function of g . The paper is organized as follows, in section 2, we introduce some notationsabout the log-concave functions. In section 3, the projection of log-concave func-tion are discussed. In section 4, we focus on how can we represent the i -th F. CHEN, J. FANG, M. LUO, C. YANG functional mixed Quermassintegrals W i ( f, g ) similar as W i ( K, L ). Owing to theBlaschke-Petkantschin formula and the similar definition of the support functionof f , we obtain the integral represent of the i -th functional mixed Quermassinte-grals W i ( f, g ). 2. preliminaries Let u : Ω → ( −∞ , + ∞ ] be a convex function, that is u (cid:0) (1 − t ) x + ty (cid:1) ≤ (1 − t ) u ( x ) + tu ( y ) for t ∈ [0 , { x ∈ R n : u ( x ) ∈ R } is the domainof u . By the convexity of u , Ω is a convex set in R n . We say that u is proper ifΩ = ∅ , and u is of class C if it is twice differentiable on int (Ω), with a positivedefinite Hessian matrix. In the following we define the subclass of u , L = (cid:8) u : Ω → ( −∞ , + ∞ ] : u is convex, low semicontinuousand lim k x k→ + ∞ u ( x ) = + ∞ (cid:9) . Recall that the Fenchel conjugate of u is the convex function defined by u ∗ ( y ) = sup x ∈ R n (cid:8) h x, y i − u ( x ) (cid:9) . (2.1)It is obvious that u ( x ) + u ∗ ( y ) ≥ h x, y i for all x, y ∈ R n , and there is an equalityif and only if x ∈ Ω and y is in the subdifferential of u at x , that means u ∗ ( ∇ u ( x )) + u ( x ) = h x, ∇ u ( x ) i . (2.2)Moreover, if u is a lower semi-continuous convex function, then also u ∗ is a lowersemi-continuous convex function, and u ∗∗ = u .The infimal convolution of functions u and v from Ω to ( −∞ , + ∞ ] defined by u (cid:3) v ( x ) = inf y ∈ Ω (cid:8) u ( x − y ) + v ( y ) (cid:9) . (2.3)The right scalar multiplication by a nonnegative real number α : (cid:0) uα (cid:1) ( x ) := (cid:26) αu (cid:0) xα (cid:1) , if α > I { } , if α = 0. (2.4)The following proposition below gathers some elementary properties of theFenchel conjugate and the infimal convolution of u and v , which can be foundin [17, 41]. Proposition 2.1.
Let u, v : Ω → ( −∞ , + ∞ ] be convex functions. Then: (1) (cid:0) u (cid:3) v (cid:1) ∗ = u ∗ + v ∗ ; (2) ( uα ) ∗ = αu ∗ , α > ; (3) dom ( u (cid:3) v ) = dom ( u ) + dom ( v ) ; (4) it holds u ∗ (0) = − inf( u ) , in particular if u is proper, then u ∗ ( y ) > −∞ ; inf( u ) > −∞ implies u ∗ is proper. The following Proposition about the Fenchel and Legendre conjugates are ob-tained in [41].
N MIXED QUERMASSINTEGRALS 5
Proposition 2.2 ( [41]) . Let u : Ω → ( −∞ , + ∞ ] be a closed convex function, andset C := int (Ω) , C ∗ := int ( dom ( u ∗ )) . Then ( C , u ) is a convex function of Legendretype if and only if C ∗ , u ∗ is. In this case ( C ∗ , u ∗ ) is the Legendre conjugate of ( C , u ) (and conversely). Moreover, ∇ u := C → C ∗ is a continuous bijection, and theinverse map of ∇ u is precisely ∇ u ∗ . A function f : R n → ( −∞ , + ∞ ] is called log-concave if for all x, y ∈ R n and 0 1, we have f (cid:0) (1 − t ) x + ty (cid:1) ≥ f − t ( x ) f t ( y ) . If f is a strictly positive log-concavefunction on R n , then there exist a convex function u : Ω → ( −∞ , + ∞ ] such that f = e − u . The log-concave function is closely related to the convex geometry of R n . An example of a log-concave function is the characteristic function χ K of aconvex body K in R n , which is defined by χ K ( x ) = e − I K ( x ) = (cid:26) , if x ∈ K ;0 , if x / ∈ K , (2.5)where I K is a lower semi-continuous convex function, and the indicator functionof K is, I K ( x ) = (cid:26) , if x ∈ K ; ∞ , if x / ∈ K . (2.6)In the later sections, we also use f to denote f been extended to R n . f = (cid:26) f, x ∈ Ω;0 , x ∈ R n / Ω. (2.7)Let A = (cid:8) f : R n → (0 , + ∞ ] : f = e − u , u ∈ L (cid:9) be the subclass of f in R n . Theaddition and multiplication by nonnegative scalars in A is defined by (see [17]). Definition 2.1. Let f = e − u , g = e − v ∈ A , and α, β ≥ 0. The sum andmultiplication of f and g is defined as α · f ⊕ β · g = e − [( uα ) (cid:3) ( vβ )] . That means (cid:0) α · f ⊕ β · g (cid:1) ( x ) = sup y ∈ R n f (cid:16) x − yα (cid:17) α g (cid:16) yβ (cid:17) β . (2.8)In particularly, when α = 0 and β > 0, we have ( α · f ⊕ β · g )( x ) = g ( xβ ) β ; when α > β = 0, then ( α · f ⊕ β · g )( x ) = f ( xα ) α ; finally, when α = β = 0, wehave (cid:0) α · f ⊕ β · g (cid:1) = I { } .The following Lemma is obtained in [17]. Lemma 2.3 ( [17]) . Let u ∈ L , then there exist constants a and b , with a > ,such that, for x ∈ Ω u ( x ) ≥ a k x k + b. (2.9) Moreover u ∗ is proper, and satisfies u ∗ ( y ) > −∞ , ∀ y ∈ Ω . The Lemma 2.3 grants that L is closed under the operations of infimal convo-lution and right scalar multiplication defined in (2.3) and (2.4) are closed. F. CHEN, J. FANG, M. LUO, C. YANG Proposition 2.4 ( [17]) . Let u and v belong both to the same class L , and α, β ≥ . Then uα (cid:3) vβ belongs to the same class as u and v . Let f ∈ A , according to papers of [5, 42], the support function of f = e − u isdefined as, h f ( x ) = ( − log f ( x )) ∗ = u ∗ ( x ) , (2.10)here the u ∗ is the Legendre transform of u . The definition of h f is a propergeneralization of the support function h K . In fact, one can easily checks h χ K = h K . Obviously, the support function h f share the most of the important propertiesof support functions h K . Specifically, it is easy to check that the function h : A →L has the following properties [43]:(1) h is a bijective map from A → L . (2) h is order preserving: f ≤ g if and only if h f ≤ h g . (3) h is additive: for every f, g ∈ A we have h f ⊕ g = h f + h g . The following proposition shows that h f is GL ( n ) covariant. Proposition 2.5 ( [18]) . Let f ∈ A . For A ∈ GL ( n ) and x ∈ R n , then h f ◦ A ( x ) = h f ( A − t x ) . (2.11)Let u, v ∈ L , denote by u t = u (cid:3) vt ( t > f t = e − u t . The followingLemmas describe the monotonous and convergence of u t and f t , respectively. Lemma 2.6 ( [17]) . Let f = e − u , g = g − v ∈ A . For t > , set u t = u (cid:3) ( vt ) and f t = e − u t . Assume that v (0) = 0 , then for every fixed x ∈ R n , u t ( x ) and f t ( x ) arerespectively pointwise decreasing and increasing with respect to t ; in particular itholds u ( x ) ≤ u t ( x ) ≤ u ( x ) and f ( x ) ≤ f t ( x ) ≤ f ( x ) ∀ x ∈ R n ∀ t ∈ [0 , . (2.12) Lemma 2.7 ( [17]) . Let u and v belong both to the same class L and, for any t > , set u t := u (cid:3) ( vt ) . Assume that v (0) = 0 , then (1) ∀ x ∈ Ω , lim t → + u t ( x ) = u ( x ) ; (2) ∀ E ⊂⊂ Ω , lim t → + ∇ u t ( x ) = ∇ u uniformly on E . Lemma 2.8 ( [17]) . Let u and v belong both to the same class L and for any t > , let u t := u (cid:3) ( vt ) . Then ∀ x ∈ int (Ω t ) , and ∀ t > , ddt (cid:0) u t ( x ) (cid:1) = − ψ ( ∇ u t ( x )) , (2.13) where ψ := v ∗ . Projection of functions onto linear subspace Let G i,n (0 ≤ i ≤ n ) be the Grassmannian manifold of i -dimensional linearsubspace of R n . The elements of G i,n will usually be denoted by ξ i and, ξ ⊥ i standsfor the orthogonal complement of ξ i which is a ( n − i )-dimensional subspace of N MIXED QUERMASSINTEGRALS 7 R n . Let ξ i ∈ G i,n and f : R n → R . The projection of f onto ξ i is defined by(see [22, 26]) f | ξ i ( x ) := max { f ( y ) : y ∈ x + ξ i ⊥ } , ∀ x ∈ Ω | ξ i . (3.1)where ξ ⊥ i is the orthogonal complement of ξ i in R n , Ω | ξ i is the projection of Ωonto ξ i . By the definition of the log-concave function f = e − u , for every x ∈ Ω | ξ i ,one can rewrite (3.1) as f | ξ i ( x ) = exp n max {− u ( y ) : y ∈ x + ξ ⊥ i } o = e − u | ξi ( x ) . (3.2)Regards the the “sum” and “multiplication” of f , we say that the projectionkeeps the structure on R n . In other words, we have the following Proposition. Proposition 3.1. Let f, g ∈ A , ξ i ∈ G i,n and α, β > . Then ( α · f ⊕ β · g ) | ξ i = α · f | ξ i ⊕ β · g | ξ i . (3.3) Proof. Let f, g ∈ A , let x , x , x ∈ ξ i such that x = αx + βx , then we have,( α · f ⊕ β · g ) | ξ i ( x ) ≥ ( α · f ⊕ β · g )( αx + βx + ξ ⊥ i ) ≥ f ( x + ξ ⊥ i ) α g ( x + ξ ⊥ i ) β . Taking the supremum of the second right hand inequality over all ξ ⊥ i we obtain( α · f ⊕ β · g ) | ξ i ≥ α · f | ξ i ⊕ β · g | ξ i . On the other hand, for x ∈ ξ i , any x , x ∈ ξ i such that x + x = x , then (cid:0) α · f | ξ i ⊕ β · g | ξ i (cid:1) ( x ) = sup x + x = x n max (cid:8) f α ( x α + ξ ⊥ i ) (cid:9) max (cid:8) g β ( x β + ξ ⊥ i ) (cid:9)o ≥ sup x + x = x n max (cid:16) f α ( x α + ξ ⊥ i ) g β ( x β + ξ ⊥ i ) (cid:17)o = max n sup x + x = x (cid:16) f α ( x α + ξ ⊥ i ) g β ( x β + ξ ⊥ i ) (cid:17)o = ( α · f ⊕ β · g ) | ξ i ( x ) . Here since f, g ≥ 0, the inequality max { f · g } ≤ max { f } · max { g } holds for x ∈ R n . So we complete the proof of the result. (cid:3) Proposition 3.2. Let ξ i ∈ G i,n , f and g are functions on R n , such that f ( x ) ≤ g ( x ) holds. Then f | ξ i ≤ g | ξ i , (3.4) holds for any x ∈ ξ i .Proof. For y ∈ x + ξ ⊥ i , since f ( y ) ≤ g ( y ), then f ( y ) ≤ max { g ( y ) : y ∈ x + ξ ⊥ i } .So, max { f ( y ) : y ∈ x + L ⊥ i } ≤ max { g ( y ) : y ∈ x + ξ ⊥ i } . By the definition of theprojection, we complete the proof. (cid:3) For the convergence of f we have the following. Proposition 3.3. Let { f i } are functions such that lim n →∞ f n = f , ξ i ∈ G i,n , then lim n →∞ ( f n | ξ i ) = f | ξ i . F. CHEN, J. FANG, M. LUO, C. YANG Proof. Since lim n →∞ f n = f , it means that, for ∀ ǫ > 0, there exist N , ∀ n > N ,such that f − ǫ ≤ f n ≤ f + ǫ . By the monotonicity of the projection, we have f | ξ i − ǫ ≤ f n | ξ i ≤ f | ξ i + ǫ . Hence each { f n | ξ i } has a convergent subsequence, wedenote it also by { f n | ξ i } , converging to some f ′ | ξ i . Then for x ∈ ξ i , we have f | ξ i ( x ) − ǫ ≤ f ′ | ξ i ( x ) = lim n →∞ ( f n | ξ i )( x ) ≤ f | ξ i ( x ) + ǫ. By the arbitrarily of ǫ we have f ′ | ξ i = f | ξ i , so we complete the proof. (cid:3) Combining with Proposition 3.3 and Proposition 2.7, it is easy to obtain thefollowing Propsosition. Proposition 3.4. Let u and v belong both to the same class L , Ω ∈ R n be thedomain of u , for any t > , set u t = u (cid:3) ( vt ) . Assume that v (0) = 0 and ξ i ∈ G i,n ,then (1) ∀ x ∈ Ω | ξ i , lim t → + u t | ξ i ( x ) = u | ξ i ( x ) , (2) ∀ x ∈ int (Ω | ξ i ) , lim t → + ∇ u t | ξ i = ∇ u | ξ i . Now let us introduce some fact about the functions u t = u (cid:3) ( vt ) with respectto the parameter t . Lemma 3.5. Let ξ i ∈ G i,n , u and v belong both to the same class L , u t := u (cid:3) ( vt ) and Ω t be the domain of u t ( t > ). Then for x ∈ Ω t | ξ i , ddt (cid:16) u t | ξ i (cid:17) ( x ) = − ψ (cid:16) ∇ (cid:0) u t | ξ i (cid:1) ( x ) (cid:17) , (3.5) where ψ := v ∗ | ξ i .Proof. Set D t := Ω t | ξ i ⊂ ξ i , for fixed x ∈ int ( D t ), the map t → ∇ (cid:0) u t | ξ i (cid:1) ( x ) isdifferentiable on (0 , + ∞ ). Indeed, by the definition of Fenchel conjugate and thedefinition of projection u , it is easy to see that ( u | ξ i ) ∗ = u ∗ | ξ i and ( u (cid:3) ut ) | ξ i = u | ξ i (cid:3) ut | ξ i hold. The Lemma 2.4 and the property of the projection grant thedifferentiability. Set ϕ := u ∗ | ξ i and ψ := v ∗ | ξ i , and ϕ t = ϕ + tψ , then ϕ t belongsto the class C on ξ i . Then ∇ ϕ t = ∇ ϕ + t ∇ ψ is nonsingular on ξ i . So theequation ∇ ϕ ( y ) + t ∇ ψ ( y ) − x = 0 , (3.6)locally defines a map y = y ( x, t ) which is of class C . By Proposition 2.2, we have ∇ ( u t | ξ i ) is the inverse map of ∇ ϕ t , that is ∇ ϕ t ( ∇ ( u t | ξ i ( x )) = x , which meansthat for every x ∈ int ( D t ) and every t > t → ∇ ( u t | ξ i ) is differentiable. Usingthe equation (2.2) again, we have u t | ξ i ( x ) = (cid:10) x, ∇ ( u t | ξ i )( x ) (cid:11) − ϕ t (cid:0) ∇ ( u t | ξ i )( x ) (cid:1) , ∀ x ∈ int ( D t ) . (3.7)Moreover, note that ϕ t = ϕ + tψ we have u t | ξ i ( x ) = D x, ∇ ( u t | ξ i )( x ) E − ϕ (cid:16) ∇ (cid:0) u t | ξ i (cid:1) ( x ) (cid:17) − tψ (cid:16) ∇ (cid:0) u t | ξ i (cid:1) ( x ) (cid:17) = u t | ξ i (cid:16) ∇ (cid:0) u t | ξ i (cid:1) ( x ) (cid:17) − tψ (cid:16) ∇ (cid:0) u t | ξ i (cid:1) ( x ) (cid:17) . N MIXED QUERMASSINTEGRALS 9 Differential the above formal we obtain, ddt (cid:16) u t | ξ i (cid:17) ( x ) = − ψ (cid:16) ∇ (cid:0) u t | ξ i (cid:1) ( x ) (cid:17) . Thenwe complete the proof of the result. (cid:3) Functional Quermassintegrals of Log-concave Function A function f ∈ A is non-degenerate and integrable if and only if lim k x k→ + ∞ u ( x ) k x k =+ ∞ . Let L ′ = (cid:8) u ∈ L : u ∈ C ( R n ) , lim k x k→ + ∞ u ( x ) k x k = + ∞ (cid:9) , and A ′ = (cid:8) f : R n → (0 , + ∞ ] : f = e − u , u ∈ L ′ (cid:9) . Now we define the i -th total mass of f . Definition 4.1. Let f ∈ A ′ , ξ i ∈ G i,n ( i = 1 , , · · · , n − x ∈ Ω | ξ i . The i -th total mass of f is defined as J i ( f ) := Z ξ i f | ξ i ( x ) dx, (4.1)where f | ξ i is the projection of f onto ξ i defined by (3.1), dx is the i -dimensionalvolume element in ξ i . Remark . (1) The definition of the J i ( f ) follows the i -dimensional volume ofthe projection a convex body. If i = 0, we defined J ( f ) := ω n , the volume of theunit ball in R n , for the completeness.(2) When take f = χ K , the characteristic function of a convex body K , onehas J i ( f ) = V i ( K ), the i -dimensional volume in ξ i . Definition 4.2. Let f ∈ A ′ . Set ξ i ∈ G i,n be a linear subspace and, for x ∈ Ω | ξ i ,the i -th functional Quermassintegrals of f (or the i -dimensional mean projectionmass of f ) are defined as W n − i ( f ) := ω n ω i Z G i,n J i ( f ) dµ ( ξ i ) , (4.2)where J i ( f ) is the i -th total mass of f defined by (4.1), dµ ( ξ i ) is the normalizedHaar measure on G i,n . Remark . (1) The definition of the W i ( f ) follows the definition of the i -thQuermassintegral W i ( K ), that is, the i -th mean total mass of f on G i,n . Alsoin the recently paper [12], the authors give the same definition by defining theQuermassintegral of the support set for the quasi-concave functions.(2) When i equals to n in (4.2), we have W ( f ) = R R n f ( x ) dx = J ( f ) , the totalmass function of f defined by Colesanti and Fragal´a [17]. Then we can say thatour definition of the W i ( f ) is a nature extension of the total mass function of J ( f ).(3) Form the definition of the Quermassintegrals W i ( f ), the following propertiesare obtained (see also [12]). • Positivity. 0 ≤ W i ( f ) ≤ + ∞ . • Monotonicity. W i ( f ) ≤ W i ( g ), if f ≤ g . • Generally speaking, the W i ( f ) has no homogeneity under dilations. Thatis W i ( λ · f ) = λ n − i W i ( f λ ) , where λ · f ( x ) = λf ( x/λ ) , λ > Definition 4.3. Let f , g ∈ A ′ , ⊕ and · denote the operations of “sum” and “mul-tiplication” in A ′ . W i ( f ) and W i ( g ) are, respectly, the i -th Quermassintegrals of f and g . Whenever the following limit exists W i ( f, g ) = 1( n − i ) lim t → + W i ( f ⊕ t · g ) − W i ( f ) t , (4.3)we denote it by W i ( f, g ), and call it as the first variation of W i at f along g , orthe i -th functional mixed Quermassintegrals of f and g . Remark . Let f = χ K and g = χ L , with K , L ∈ K n . In this case W i ( f ⊕ t · g ) = W i ( K + tL ), then W i ( f, g ) = W i ( K, L ). In general, W i ( f, g ) has no analogproperties of W i ( K, L ), for example, W i ( f, g ) is not always nonnegative and finite.The following is devote to prove that W i ( f, g ) exist under the fairly weak hy-pothesis. First, we prove that the first i -dimensional total mass of f is translationinvariant. Lemma 4.4. Let ξ i ∈ G i,n , f = e − u , g = e − v ∈ A ′ . Let c = inf u | ξ i =: u (0) , d = inf v | ξ i := v (0) , and set e u i ( x ) = u | ξ i ( x ) − c , e v i ( x ) = v | ξ i ( x ) − d , e ϕ i ( y ) =( e u i ) ∗ ( y ) , e ψ i ( y ) = ( e v i ) ∗ ( y ) , e f i = e − e u i , e g i = e − e v i , and e f t | i = e f ⊕ t · e g . Then if lim t → + J i ( e f t ) − J i ( e f ) t = R ξ i e ψ i dµ i ( e f ) holds, then we have lim t → + J i ( f t ) − J i ( f ) t = R ξ i ψ i dµ i ( f ) . Proof. By the construction, we have e u i (0) = 0 , e v i (0) = 0 , and e v i ≥ , e ϕ i ≥ , e ψ i ≥ 0. Further, e ψ i ( y ) = ψ i ( y ) + d , and e f i = e c f i . Solim t → + J i ( e f t ) − J i ( e f ) t = Z ξ i e ψ i dµ i ( e f ) = e c Z ξ i ψ i dµ i ( f ) + de c Z ξ i dµ i ( f ) . (4.4)On the other hand, since f i ⊕ t · g i = e − ( c + dt ) ( e f i ⊕ t · e g i ) , we have, J i ( f ⊕ t · g ) = e − ( c + dt ) J i ( e f i ⊕ t · e g i ) . Derivation both sides of the above formula, we obtainlim t → + J i ( f ⊕ t · g ) − J i ( f ) t = − de − c lim t → + J i ( e f i ⊕ t e g i ) dx + e − c lim t → + h J i ( e f t ) − J i ( e f ) t i = − de − c J i ( e f i ) + Z ξ i ψ i dµ i ( f ) + d Z ξ i dµ i ( f )= Z ξ i ψ i dµ i ( f ) . So we complete the proof. (cid:3) Theorem 4.5. Let f, g ∈ A ′ , and satisfy −∞ ≤ inf(log g ) ≤ + ∞ and W i ( f ) > . Then W j ( f, g ) is defferentiable at f along g , and it holds W j ( f, g ) ∈ [ − k, + ∞ ] , (4.5) where k = max { d, } W i ( f ) .Proof. Let ξ i ∈ G i,n , since u | ξ i := − log( f | ξ i ) = − (log f ) | ξ i and v | ξ i := − log( g | ξ i ) = − (log f ) | ξ i . By the definition of f t and the Proposition 3.1 we obtain, f t | ξ i = ( f ⊕ t · g ) | ξ i = f | ξ i ⊕ t · g | ξ i . Notice that v | ξ i (0) = v (0), set d := v (0), e v | ξ i ( x ) := v | ξ i ( x ) − d , N MIXED QUERMASSINTEGRALS 11 e g | ξ i ( x ) := e − e v | ξi ( x ) , and e f t | ξ i := f | ξ i ⊕ t · e g | ξ i . Up to a translation of coordinates,we may assume inf( v ) = v (0) . The Lemma 2.6 says that for every x ∈ ξ i , f | ξ i ≤ e f t | ξ i ≤ e f | ξ i , ∀ x ∈ R n , ∀ t ∈ [0 , . Then there exists e f | ξ i ( x ) := lim t → + e f t | ξ i ( x ). Moreover, it holds e f | ξ i ( x ) ≥ f | ξ i ( x ) and e f t | ξ i is pointwise decreasing as t → + . Lemma 2.3 and Proposition 2.4 show that f | ξ i ⊕ t · e g | ξ i ∈ A ′ , ∀ t ∈ [0 , . Then J i ( f ) ≤ J i ( e f t ) ≤ J i ( e f ), −∞ ≤ J i ( f ) , J i ( e f ) < ∞ . Hence, by the monotone and convergence, we have lim t → + W i ( e f t ) = W i ( e f ) . In fact, by definition we have e f t | ξ i ( x ) = e − inf { u | ξi ( x − y )+ tv | ξi ( yt ) } , and − inf { u | ξ i ( x − y ) + tv | ξ i ( yt ) } ≤ − inf u | ξ i ( x − y ) − t inf v | ξ i ( yt ) , Note that −∞ ≤ inf( v | ξ i ) ≤ + ∞ , then − inf u | ξ i ( x − y ) − t inf v | ξ i ( yt ) is continuousfunction of variable t , then e f | ξ i ( x ) := lim t → + e f t | ξ i ( x ) = f | ξ i ( x ) . (4.6)Moreover, W i ( e f t ) is continuous function of t ( t ∈ [0 , t → + W i ( e f t ) = W i ( f ) . Since f t | ξ i = e − dt e f | ξ i ( x ), we have W i ( f t ) − W i ( f ) t = W i ( f ) e − dt − t + e − dt W i ( e f t ) − W i ( f ) t . (4.7)Notice that, e f t | ξ i ≥ f | ξ i , we have the following two cases, that is: ∃ t > W i ( e f t ) = W i ( f ) or W i ( e f t ) = W i ( f ), ∀ t > . For the first case, since W i ( e f t ) is a monotone increasing function of t , it mustholds W i ( e f t ) = W i ( f ) for every t ∈ [0 , t ]. Hence we have lim t → + W i ( f t ) − W i ( f ) t = − dW i ( f ), the statement of the theorem holds true.In the latter case, since e f t | ξ i is increasing non-negative function, it means thatlog( W i ( e f t )) is an increasing concave function of t . Then, ∃ log( W i ( e f t )) − log( W i ( f )) t ∈ [0 , + ∞ ] . On the other hand, sincelog ′ (cid:16) W i ( e f t ) (cid:17)(cid:12)(cid:12)(cid:12) t =0 = lim t → + log( W i ( e f t )) − log( W i ( f )) W i ( e f t ) − W i ( f ) = 1 W i ( f ) . Then lim t → + W i ( e f t ) − W i ( f )log( W i ( e f t )) − log( W i ( f )) = W i ( f ) > . (4.8)From above we infer that ∃ lim t → + W i ( e f t ) − W i ( f ) t ∈ [0 , + ∞ ] . Combining the aboveformulas we obtainlim t → + W i ( f t ) − W i ( f ) t ∈ [ − max { d, } W i ( f ) , + ∞ ] . So we complete the proof. (cid:3) In view of the example of the mixed Quermassintegral, it is natural to askwhether in general, W i ( f, g ) has some kind of integral representation. Definition 4.4. Let ξ i ∈ G i,n and f = e − u ∈ A ′ . Consider the gradient map ∇ u : R n → R n , the Borel measure µ i ( f ) on ξ i is defined by µ i ( f ) := ( ∇ u | ξ i ) ♯ k x k n − i ( f | ξ i ) . Recall that the following Blaschke-Petkantschin formula is useful. Proposition 4.6 ( [25]) . Let ξ i ∈ G i,n ( i = 1 , , · · · , n ) be linear subspace of R n , f be a non-negative bounded Borel function on R n , then Z R n f ( x ) dx = ω n ω i Z G i,n Z ξ i f ( x ) k x k n − i dxdµ ( ξ i ) . (4.9)Now we give a proof of Theorem 1.1. Proof of Theorem 1.1. By the definition of the i -th Quermassintegral of f , wehave W i ( f t ) − W i ( f ) t = ω n ω n − i Z G n − i,n J n − i ( f t ) − J n − i ( f ) t dµ ( ξ n − i ) . Let t > C ⊂⊂ Ω | ξ n − i , and by reduction 0 ∈ int (Ω) | ξ n − i , we have C ⊂⊂ Ω | ξ n − i , by Lemma 3.5, we obtainlim h → J n − i ( f t + h )( x ) − J n − i ( f t ( x )) h = Z ξ n − i ψ (cid:0) ∇ u t | ξ n − i ( x ) (cid:1) f t | ξ n − i ( x ) dx, where ψ = h g | ξn − i = v | ∗ ξ n − i . Then we have,lim h → W i ( f t + h ) − W i ( f t ) h = ω n ω n − i Z G n − i,n Z ξ n − i ψ (cid:0) ∇ u t | ξ n − i ( x ) (cid:1) f t | ξ n − i ( x ) k x k n − i k x k n − i dxdµ ( ξ n − i ) , = Z R n ψ (cid:0) ∇ u t | ξ n − i ( x ) (cid:1) f t | ξ n − i ( x ) k x k n − i dx = Z R n ψdµ n − i ( f t ) . So we have, W i ( f t + h ) − W i ( f t ) = R t n R R n ψdµ n − i ( f s ) o ds. The continuous of ψ implies lim s → + R R n ψdµ n − i ( f s ) ds = R R n ψdµ n − i ( f ) ds. Therefore,lim t → + W i ( f t ) − W i ( f ) t = ddt W i ( f t ) | t =0 + = lim s → + ddt W i ( f t ) | t = s = lim s → + ddt Z t n Z R n ψdµ n − i ( f s ) o ds = Z R n ψdµ n − i ( f ) . N MIXED QUERMASSINTEGRALS 13 Since ψ = h g | ξ , so we have W i ( f, g ) = 1 n − i lim t → + W i ( f t ) − W i ( f ) t = 1 n − i Z R n h g | ξn − i dµ n − i ( f ) . So we complete the proof. (cid:3) Remark . From the integral representation (1.8), the i -th functional mixedQuermassintegral is linear in its second argument, with the sum in A ′ , for f, g, h ∈A ′ , then we have W i ( f, g ⊕ h ) = W i ( f, g ) + W i ( f, h ) . Data Availability: No data were used to support this study. Conflicts of Interest: The authors declare no conflict of interest. Authors Contributions: All authors contributed equally to this work. Allauthors have read and agreed to the published version of this manuscript. Acknowledgments The authors would like to strongly thank the anonymous referee for the veryvaluable comments and helpful suggestions that directly lead to improve the orig-inal manuscript. References [1] A. Aleksandrov, On the theory of mixed volumes. 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School of Mathematics and Statistics, Guizhou University of Finance andEconomics, Guiyang, Guizhou 550025, People’s Republic of China. E-mail address : [email protected] E-mail address : 2. School of Mathematical Sciences, Guizhou Normal University, Guiyang,Guizhou 550025, People’s Republic of China E-mail address : [email protected] E-mail address ::